Set \(v_i=\log _px_i\) ; that is, \(t\mapsto \exp _p(t\cdot v_i)\)  for \(t\in [0,1]\)  is a minimizing geodesic from \(p\)  to \(x_i\)  or, equivalently, \(|v_i|=|p-x_i|\)  and \(\exp _pv_i=x_i\) .
Recall that the gradient exponent \(_p\:p\rightarrow M\)  is defined for any compete Riemannian manifold;
it is is a short map if \(M\)  has nonnegative curvature
and \(_p\:v_i\mapsto x_i\)  for each \(i\)  (see [1]}).
